We propose a general framework to solve tight binding models in D dimensional lattices driven by ac electric fields. Our method is valid for arbitrary driving regimes and allows to obtain effective Hamiltonians for different external fields configurations. We establish an equivalence with time independent lattices in D+1 dimensions, and analyze their topological properties. Further, we demonstrate that non-adiabaticity drives a transition from topological invariants defined in D+1 to D dimensions. Our approach provides a theoretical framework to analyze ac driven systems, with potential applications in topological states of matter, and non-adiabatic topological quantum computation, predicting novel outcomes for future experiments.Introduction: Periodically driven quantum systems has been a fastly growing research field in the last years. The development of effective Hamiltonians describing ac driven systems at certain regimes, has allowed to predict novel properties such as topological phases [1][2][3], and quantum phase transitions [4,5] that otherwise, would be impossible to achieve in the undriven case. Therefore, the application of ac fields has become a very promising tool to engineer quantum systems. On the other hand, the obtention of effective Hamiltonians can be a difficult task, depending on the driving regime to be considered.In this work, we provide a general framework to study periodically driven quantum lattices. By means of our approach, it is possible to solve with arbitrary accuracy their time evolution, and obtain effective Hamiltonians for the different driving regimes. We consider solutions of Floquet-Bloch form, based on the symmetries of the system, and characterize the states in terms of the quasienergies, which are well defined for all driving regimes. The states belong to a composed Hilbert space, in which time is treated as a parameter. As we will see below, it allows to formally describe the ac driven D dimensional lattice, as analog to a time independent D+1 dimensional lattice. This description enlightens the underlaying structure of periodically driven systems, in which the initial Bloch band splits into several copies (FloquetBloch bands), where the coupling between them directly depends on the driving regime. Interestingly, the isolated Floquet-Bloch bands possess the same topological properties as the Bloch bands of the undriven system, now tuned by the external field parameters. Thus, the topological invariants for the isolated Floquet-Bloch bands can be obtained following the general classification of time independent systems (AZ classes [6][7][8]). However, we also demonstrate that lowering the frequency, the bands couple to each other. In that case, the topological structures are classified according to a base manifold of dimension D+1. In consequence, one can simulate higher dimensional tight binding (TB) models with exotic tunable hoppings by just coupling the system to ac electric fields.Our approach is valid for arbitrary dimension, and it

We show theoretically that periodically driven systems with short range Hubbard interactions offer a feasible platform to experimentally realize fractional Chern insulator states. We exemplify the procedure for both the driven honeycomb and the square lattice, where we derive the effective steady state band structure of the driven system by using the Floquet theory and subsequently study the interacting system with exact numerical diagonalization. The fractional Chern insulator state equivalent to the 1/3 Laughlin state appears at 7/12 total filling (1/6 filling of the upper band). The state also features spontaneous ferromagnetism and is thus an example of the spontaneous breaking of a continuous symmetry along with a topological phase transition. We discuss light-driven graphene and shaken optical lattices as possible experimental systems that can realize such a state.

We investigate the effect of an in-plane ac electric field coupled to electrons in the honeycomb lattice and show that it can be used to manipulate the Dirac points of the electronic structure. We find that the position of the Dirac points can be controlled by the amplitude and the polarization of the field for high-frequency drivings, providing a new platform to achieve their merging, a topological transition which has not been observed yet in electronic systems. Importantly, for lower frequencies we find that the multiphoton absorptions and emissions processes yield the creation of additional pairs of Dirac points. This provides an additional method to achieve the merging transition by just tuning the frequency of the driving. Our approach, based on Floquet formalism, is neither restricted to specific choice of amplitude or polarization of the field, nor to a low-energy approximation for the Hamiltonian.

We extend the standard SSH model to include long-range hopping amplitudes and disorder, and analyze how the electronic and topological properties are affected. We show that long-range hoppings can change the symmetry class and the topological invariant, while diagonal and off-diagonal disorder lead to Anderson localization. Interestingly we find that the Lyapunov exponent γ(E) can be linked in two ways to the topological properties in the presence of disorder: Either due to the different response of mid-gap states to increasing disorder, or due to an extra contribution to γ due to the presence of edge modes. Finally we discuss its implications in realistic transport measurements.where H R acts on the pseudospinor Ψ k = (a k , b k ) T and p ranges from 1 to (R + 1)/2 ( · · · denotes the floor function). The dispersion relation is E ± (k) = d 0 (k) ± | d(k)|, where "+" and "−" correspond to the conduction and valence band, respectively (see Fig. 1,(b)). arXiv:1810.02779v2 [cond-mat.mes-hall]

Floquet Majorana Fermions appear as steady states at the boundary of time-periodic topological phases of matter. In this work, we theoretically study the main features of these exotic topological phases in the periodically driven one-dimensional Kitaev model. By controlling the ac fields, we can predict new topological phase transitions that should give rise to signatures of Majorana states in experiments. Moreover, the knowledge of the time-dependence of these Majorana states allows one to manipulate them. Our work contains a complete analysis of the monochromatic driving in different frequency regimes.